Lemma 1 [Azuma's Inequality [1]}]
Let \(\Sigma _0, \Sigma _1,...\)  be a martingale with respect to filtration \(\mathcal {F}_0,\mathcal {F}_1,...\)  such that \(|\Sigma _i-\Sigma _{i-1}|\le c\)  almost surely. Then for all \(\epsilon >0\) , we have that
\(\mathbb {P}[|\Sigma _n-\Sigma _0|>\epsilon ]\le 2\exp (-\frac{\epsilon ^2}{2nc^2}).\) 
